Ref. [1]} considered such constraint in the context of the
\(Z_{\cal N}\)  axion model of Ref. [2]}.
We here revisit the
latter analysis with
the formulae derived in Ref. [3]} for the \(Z_{\cal N}\)  scenario under discussion.
For the latter, the potential in the large \({\cal N}\)  limit –Eq: fourier potential large N hyper,maZNLargeN– corresponds to a quartic coupling given by
\(\lambda _a = m_a^2 f_a^2 / {\cal N}^2\,,\) 
